On the Global Evolution of Self-Gravitating Matter. Nonlinear Interactions in Gowdy Symmetry

Abstract

We are interested in the evolution of a compressible fluid under its self-generated gravitational field. Assuming here Gowdy symmetry, we investigate the algebraic structure of the Euler equations satisfied by the mass density and velocity field. We exhibit several interaction functionals that provide us with a uniform control on weak solutions in suitable Sobolev norms or in bounded variation. These functionals allow us to study the local regularity and nonlinear stability properties of weakly regular fluid flows governed by the Euler–Gowdy system. In particular, for the Gowdy equations, we prove that a spurious matter field arises under weak convergence, and we establish the nonlinear stability of weak solutions.

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Acknowledgments

Both authors gratefully acknowledge support from the Simons Center for Geometry and Physics, Stony Brook University, at which some of the research for this paper was performed. The second author (PLF) was partially supported by the Innovative Training Network (ITN) Grant 642768 (entitled ModCompShock) and by the Centre National de la Recherche Scientifique (CNRS). This paper was completed when the second author was a visiting fellow at the Courant Institute for Mathematical Sciences, New York University.